OFFSET
1,1
COMMENTS
The equation in Theorem 3.2 of the paper contains a typographical index error: The correct denominator in the e.g.f. is phi_0(x)^2-phi_1(x)*phi_3(x), equivalent to eq. (3.14).
LINKS
L. Carlitz and Richard Scoville, Enumeration of rises and falls by position, Discr. Math. 5 (1973) 45-59, variable B(4n-1).
MAPLE
Ph := proc(i, x)
local n;
add( x^(4*n+i)/(4*n+i)!, n=0..90) ;
end proc:
g := (Ph(1, x)*Ph(2, x)-Ph(0, x)*Ph(3, x) ) / (Ph(0, x)^2-Ph(1, x)*Ph(3, x)) ;
for n from 3 to 29 by 4 do
coeftayl(g, x=0, n)*n! ;
end do;
MATHEMATICA
Ph[i_, x_] = Sum[x^(4*n + i)/(4*n + i)!, {n, 0, Infinity}];
g = (Ph[1, x]*Ph[2, x]-Ph[0, x]*Ph[3, x]) / (Ph[0, x]^2-Ph[1, x]*Ph[3, x]);
a[n_] := SeriesCoefficient[g, {x, 0, 4 n - 1}]*(4 n - 1)!;
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, May 28 2024, after R. J. Mathar *)
CROSSREFS
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jul 17 2012
STATUS
approved